Answer
2 and 5 are the eigenvalues of A.
Work Step by Step
1. Calculate $Av_1$ and $Av_2$:
$$ Av_1 =
\begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}
\begin{bmatrix} 1 \\ -2 \end{bmatrix} =
\begin{bmatrix} 2 \\ -4 \end{bmatrix}
$$
$$ Av_2 =
\begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}
\begin{bmatrix} 1 \\ 1 \end{bmatrix} =
\begin{bmatrix} 5 \\ 5 \end{bmatrix}
$$
2. Find which number should multiply $v_1$ to result in $Av_1$ and the same for $v_2$ and $Av_2$
$$\begin{bmatrix} 2 \\ -4 \end{bmatrix} = 2 \begin{bmatrix} 1 \\ -2 \end{bmatrix} $$
$$
\begin{bmatrix} 5 \\ 5 \end{bmatrix} = 5\begin{bmatrix} 1 \\ 1 \end{bmatrix}
$$
Hint: Use the division operation between the numbers on the same position on the vectors.
3. Thus, 2 and 5 are the eigenvalues of A.
